Spectra of finitely generated Boolean flows

John F. Kennison

A flow on a compact Hausdorff space X is given by a map
t : X --> X. The general goal of this paper is to find the "cyclic
parts" of such a flow. To do this, we approximate (X,t) by a flow
on a Stone space (that is, a totally disconnected, compact Hausdorff
space). Such a flow can be examined by analyzing the resulting flow
on the Boolean algebra of clopen subsets, using the spectrum defined
in our previous TAC paper, The cyclic spectrum of a Boolean flow.
In this paper, we describe the cyclic spectrum in terms that do not
rely on topos theory. We then compute the cyclic spectrum of any
finitely generated Boolean flow. We define when a sheaf of Boolean
flows can be regarded as cyclic and find necessary conditions for
representing a Boolean flow using the global sections of such a
sheaf. In the final section, we define and explore a related
spectrum based on minimal subflows of Stone spaces.